Parallel imaging techniques are applied in magnetic resonance imaging (MRI) to improve spatial resolution, temporal sampling, and/or signal to noise ratio (SNR). GRAPPA (GeneRalized Auto-calibrating Partially Parallel Acquisitions) is one parallel imaging reconstruction technique. In GRAPPA, a plurality of ACS (auto calibration signal) lines are acquired in addition to normal down-sampled data. These ACS lines may be used in final reconstruction. However, these ACS lines may also be used for calculating coil coefficients that are used to fill k-space lines that are missing due to partial acquisition.
GRAPPA may employ a least-squares technique to compute the coil coefficients that are used to fill the missing k-space lines. In one example, computing the coil coefficients may include solving over determined equations. Due to the nature of the over determined equations, both estimation accuracy and reconstructed image quality may be negatively affected by outliers in the acquired data.
Conventional GRAPPA may be applied to data acquired using for example, Cartesian sampling trajectories. The acquired k-space data may include two portions: ACS lines from the center region of k-space, and under-sampled outer k-space data. The ACS lines are sampled at the Nyquist rate. However, the under-sampled outer k-space data may be sampled at a lower rate described by reducing the Nyquist rate by an amount referred to as the outer reduction factor (ORF). Additionally, k-space signals may be acquired at a first sampling density in the read direction and at a second more sparse sampling rate in the phase encoding (PE) direction. Additional ACS lines may be acquired near the center of k-space to facilitate calculating coil coefficients.
Prior Art FIG. 1 illustrates one example reconstruction method 100 that includes two logical portions and that performs GRAPPA. A first portion (120-130) follows k-space data acquisition at 110. The k-space data may include both ACS lines and sub-sampled outer k-space data. Method 100 includes, at 120, includes creating over determined linear equations based, at least in part, the ACS lines acquired at 110. Coil coefficients are then estimated at 130 by solving the equations using, for example, a least-squares fit. Blocks of signals from one and/or multiple coils may be used to fit a single ACS line in one coil. A block may include one line of measured signal and (ORF−1) lines of missing signal.
In one example, the least squares fit process may be described by linear equations having the form:
                                          S            j            ACS                    ⁡                      (                                          k                y                            +                              m                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                                  k                  y                                                      )                          =                              ∑                          l              =              1                        L                    ⁢                                    ∑                              b                =                1                            4                        ⁢                                          n                                  j                  ,                  h                  ,                  l                                m                            ⁢                                                S                  l                                ⁡                                  (                                                            k                      y                                        +                                          bORF                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                                              k                        y                                                                              )                                                                                        Equation        ⁢                                  ⁢        1            
where j is the index of the target coil (j=1, 2, . . . L), l is the index of the L coils, b is the index of the block (b=1-4), n refers to the coil coefficients, and S refers to the measured signals. The coil coefficients n may be obtained using a pseudo-inverse matrix operation. Equation (1) may be applied to each of the L target coils.
A second portion of the prior art method 100 includes, at 140, determining the missing k-space data in outer k-space. This may be achieved by applying equation (1) at different k-space locations using the coil coefficients established at 130 and the sub-sampled k-space data acquired at 110. The missing k-space lines can be reconstructed from this data using direct matrix multiplication. The final full k-space data are created at 150 by combining the k-space data estimated at 140 and the actual k-space data acquired at 110. An image can then be computed at 160 from the final full k-space data created at 150 using conventional techniques. Thus it can be seen that GRAPPA considers every data point equally, which may not be necessary, may lead to artifacts, and may unnecessarily increase processing time.